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			<h1 id="firstHeading" class="firstHeading">Upsampling</h1>
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				<p><b>Upsampling</b> is the process of <a href="http://en.wikipedia.org/wiki/Sample_rate_conversion" title="Sample rate conversion">increasing the sampling rate</a> of a <a href="http://en.wikipedia.org/wiki/Signal_%28information_theory%29" title="Signal (information theory)" class="mw-redirect">signal</a>. For instance, upsampling raster images such as photographs means increasing the resolution of the image.</p>
<p>The upsampling factor (commonly denoted by <i>L</i>) is usually an integer or a rational fraction greater than unity. This factor multiplies the <a href="http://en.wikipedia.org/wiki/Sampling_rate">sampling rate</a> or, equivalently, divides the sampling period. For example, if <a href="http://en.wikipedia.org/wiki/Compact_disc" class="mw-redirect" title="Compact disc">compact disc</a> audio is upsampled by a factor of 5/4 then the resulting sampling rate goes from 44,100 <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz">Hz</a> to 55,125 Hz.</p>
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<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Filtering"><span class="tocnumber">1</span> <span class="toctext">Filtering</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Upsampling_process"><span class="tocnumber">2</span> <span class="toctext">Upsampling process</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Upsampling_by_integer_factor"><span class="tocnumber">2.1</span> <span class="toctext">Upsampling by integer factor</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Upsampling_by_rational_fraction"><span class="tocnumber">2.2</span> <span class="toctext">Upsampling by rational fraction</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-5"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#References"><span class="tocnumber">4</span> <span class="toctext">References</span></a></li>
</ul>
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<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Upsampling&amp;action=edit&amp;section=1" title="Edit section: Filtering">edit</a>]</span> <span class="mw-headline" id="Filtering">Filtering</span></h2>
<p>For an aesthetically pleasing upsample, an <a href="http://en.wikipedia.org/wiki/Interpolation">interpolation</a> filter is required; in both upsampling and <a href="http://en.wikipedia.org/wiki/Downsampling">downsampling</a>, such a <a href="http://en.wikipedia.org/wiki/Low-pass_filter">low-pass filter</a> implements <a href="http://en.wikipedia.org/wiki/Anti-aliasing" class="mw-redirect" title="Anti-aliasing">anti-aliasing</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Upsampling&amp;action=edit&amp;section=2" title="Edit section: Upsampling process">edit</a>]</span> <span class="mw-headline" id="Upsampling_process">Upsampling process</span></h2>
<p>Consider a <a href="http://en.wikipedia.org/wiki/Discrete_signal">discrete signal</a> <span class="texhtml"><i>f</i>(<i>k</i>)</span> on a radian frequency <a href="http://en.wikipedia.org/wiki/Digital_frequency">digital frequency</a> range.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Upsampling&amp;action=edit&amp;section=3" title="Edit section: Upsampling by integer factor">edit</a>]</span> <span class="mw-headline" id="Upsampling_by_integer_factor">Upsampling by integer factor</span></h3>
<p>Let <i>L</i> denote the upsampling factor.</p>
<ol>
<li>Add <i>L-1</i> zeros between each sample in <span class="texhtml"><i>f</i>(<i>k</i>)</span>. Or, equivalently define <img class="tex" alt="g(k) = \left \{ \begin{matrix} f\left(\frac{k}{L}\right) &amp; \mbox{if } \frac{k}{L} \mbox{ is an integer} \\ 0 &amp; \mbox{otherwise} \end{matrix} \right." src="wikipedia-Upsampling_pliki/85366f39c8a2a4a9a3ab930c407cbb3d.png"></li>
<li>Filter with a low-pass filter which, theoretically, should be the <a href="http://en.wikipedia.org/wiki/Sinc_filter">sinc filter</a> with frequency cut off at <img class="tex" alt="\frac{\pi}{L}" src="wikipedia-Upsampling_pliki/e76ff7b94a2926e073bd2c413e11639f.png"></li>
</ol>
<p>The second step calls for the use of a perfect low-pass filter, which
 is not implementable. When choosing a realizable low-pass filter this 
will have to be considered and it will have <a href="http://en.wikipedia.org/wiki/Aliasing">aliasing</a>
 effects. These aliases can be removed to a reasonable extent by a 
finite impulse response low pass filter. The presence of zeros in the 
sequence which is passed through the filter can be used to reduce the 
complexity of the filter implementation. The original filter can be 
split to <span class="texhtml"><i>L</i></span> subfilters and the output of each of these subfilters is sequentially tapped to obtain the filtered output sequence.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Upsampling&amp;action=edit&amp;section=4" title="Edit section: Upsampling by rational fraction">edit</a>]</span> <span class="mw-headline" id="Upsampling_by_rational_fraction">Upsampling by rational fraction</span></h3>
<p>Let <i>L/M</i> denote the upsampling factor.</p>
<ol>
<li>Upsample by a factor of <i>L</i></li>
<li><a href="http://en.wikipedia.org/wiki/Downsampling" title="Downsampling">Downsample</a> by a factor of <i>M</i></li>
</ol>
<p>Note that upsampling requires an interpolation filter after 
increasing the data rate and that downsampling requires a filter before 
decimation. These two filters can be combined into a single filter. 
Since both interpolation and anti-aliasing filters are low-pass filters,
 the filter with the smallest bandwidth is more restrictive and, thus, 
can be used in place of both filters. Since the rational fraction <i>L/M</i> is greater than unity when <span class="texhtml"><i>M</i> &lt; <i>L</i></span>, the single low-pass filter should have cutoff at <span class="texhtml">1 / 2<i>L</i></span> cycles per intermediate sample, the <a href="http://en.wikipedia.org/wiki/Nyquist_frequency">Nyquist frequency</a> of the input sample rate.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Upsampling&amp;action=edit&amp;section=5" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Downsampling">Downsampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Oversampling">Oversampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sampling_%28information_theory%29" class="mw-redirect" title="Sampling (information theory)">Sampling (information theory)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Signal_%28information_theory%29" class="mw-redirect" title="Signal (information theory)">Signal (information theory)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Data_conversion">Data conversion</a></li>
<li><a href="http://en.wikipedia.org/wiki/Interpolation">Interpolation</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Upsampling&amp;action=edit&amp;section=6" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<ul>
<li><span class="citation book"><a href="http://en.wikipedia.org/wiki/Alan_V_Oppenheim" title="Alan V Oppenheim" class="mw-redirect">Oppenheim, Alan V.</a>; Ronald W. Schafer, John R. Buck (1999). <i>Discrete-Time Signal Processing</i> (2nd Edition ed.). Prentice Hall. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-13-754920-2" title="Special:BookSources/0-13-754920-2">0-13-754920-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Discrete-Time+Signal+Processing&amp;rft.aulast=Oppenheim&amp;rft.aufirst=Alan+V.&amp;rft.au=Oppenheim%2C%26%2332%3BAlan+V.&amp;rft.date=1999&amp;rft.edition=2nd+Edition&amp;rft.pub=Prentice+Hall&amp;rft.isbn=0-13-754920-2&amp;rfr_id=info:sid/en.wikipedia.org:Upsampling"><span style="display: none;">&nbsp;</span></span></li>
</ul>
<ul>
<li><a href="http://www.eetimes.com/design/automotive-design/4017637/Multirate-DSP-part-1-Upsampling-and-downsampling" class="external text" rel="nofollow">Upsampling and downsampling</a></li>
</ul>
<ul>
<li><a href="http://ccrma.stanford.edu/%7Ejos/resample/resample.html" class="external text" rel="nofollow">Digital Audio Resampling Home Page</a> (discusses a technique for bandlimited interpolation)</li>
</ul>
<ul>
<li>Matlab example of using polyphase filters for interpolation <a href="http://www.dsplog.com/2007/05/12/polyphase-filters-for-interpolation/" class="external autonumber" rel="nofollow">[1]</a></li>
</ul>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete-time_signal" class="mw-redirect" title="Discrete-time signal">discrete-time signal</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Sampling_theorem" class="mw-redirect" title="Sampling theorem">sampling theorem</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Estimation_theory">estimation theory</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Detection_theory">detection theory</a></span></div>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Audio_signal_processing">audio signal processing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Control_engineering">control engineering</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Digital_image_processing">digital image processing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Speech_processing">speech processing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Statistical_signal_processing">statistical signal processing</a></span></div>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform">Discrete Fourier transform</a> (DFT)&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform">Discrete-time Fourier transform</a> (DTFT)&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Impulse_invariance">Impulse invariance</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Bilinear_transform">bilinear transform</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Pole%E2%80%93zero_mapping" class="mw-redirect" title="Pole–zero mapping">pole–zero mapping</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Z-transform">Z-transform</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Advanced_Z-transform">advanced Z-transform</a></span></div>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling</a></td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Oversampling">oversampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Undersampling">undersampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Downsampling">downsampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><strong class="selflink">upsampling</strong>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Aliasing">aliasing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter">anti-aliasing filter</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Sampling_rate">sampling rate</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Nyquist_rate">Nyquist rate</a>/<a href="http://en.wikipedia.org/wiki/Nyquist_frequency" title="Nyquist frequency">frequency</a></span></div>
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